S
ONDER
F
ORSCHUNGS
B
EREICH
504
Rationalit ¨atskonzepte,
Entscheidungsverhalten und
¨okonomische Modellierung
Universit ¨
at Mannheim
L 13,15
68131 Mannheim
No. 03-32
Reciprocity - an indirect evolutionary analysis
Siegfried K. Berninghaus
∗and Christian Korth
∗∗and Stefan Napel
∗∗∗October 2003
Financial support from the Deutsche Forschungsgemeinschaft, SFB 504, at the University of Mannheim, is gratefully acknowledged.
∗Universit ¨at Karlsruhe, email: [email protected] ∗∗, email: [email protected]
Reciprocity – An Indirect Evolutionary Analysis
Siegfried K. Berninghaus Christian Korth
School of Economics Niersstrasse 12 and Business Engineering D-47533 Kleve
University of Karlsruhe Germany D-76128 Karlsruhe Fax: +49-2821-583748
Germany [email protected] Fax: +49-721-608-3491
Stefan Napel1 University of Hamburg and
Boston University Department of Economics Boston, MA 02215 USA Fax: +1-617-353-4449 [email protected] Version: September 29, 2003
This paper investigates strategic interaction between rational agents whose preferences evolve over time. Players face a pecuniary ‘game of life’ comprising the ultimatum game and the dictator game. Utility may but need not be attached to the reciprocation of fair and unfair play by the opponent and equitable payoff distributions as proposed by Falk and Fischbacher (2001). Evolutionary fitness is determined solely by material success – regardless of the motives for its achievement. Agents cannot explicitly condition the social component of their preferences on whether they face the ultimatum or dictator game. Under these conditions, agents develop a strong preference for reciprocation but little interest in an equitable distribution as such. This corresponds to equitable ultimatum offers but full surplus appropriation by dictators. Adding an exogenous constraint on the possible divergence between preference for reciprocation and for an equitable distribution either makes ultimatum divisions asymmetric or dictators become generous depending on the relative frequency of ultimatum and dictator interaction.
1Financial support from the German Academic Exchange Service (DAAD) is gratefully
1. INTRODUCTION
In experimental investigations of the ultimatum game, participants quite con-sistently offer 30-50% of an available monetary surplus as first-moving proposers. They reject offers of less than 20% as second-moving responders, which results in no player receiving anything. Particularly the latter observation is hard to reconcile with the assumption that economic actors are rational maximizers of their mone-tary payoffs.2 By including a consideration for fairness conventions and reciprocity
in players’ preferences observations can be explained very well. This is true not only for the ultimatum game, but also many other games for which experimental findings are puzzling from a monetary-payoff maximization point of view.
Neoclassical theory does not restrict preferences to be based only on monetary payoffs or to be strictly monotonic in them. But economic agents who are spiteful, enjoy a warm glow donating money to anonymous strangers, or feel it worthwhile to incur private costs to punish free-riders of public goods have certainly not been the conventional assumption in economics. The award of 2002’s Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel to Daniel Kahneman and Vernon Smith for “Foundations of Behavioral and Experimental Economics” is only one indicator that this is changing fast.
In adopting a more realistic view of homo oeconomicus, however, one should be careful to not jump to the other extreme, i. .e. to count any observation of supposedly odd behavior as evidence that ‘standard assumptions’ are wrong and to suppose that human beings universally exhibit the nicer-than-expected character exhibited in some laboratory experiments. The latter would be invalidated by many other experiments, e. g. on market games, in which participants’ behavior is explained well by egoistic maximization of monetary rewards. Also, the tale – told in different versions before the advent of behavioral economics – that (an unspecified kind of) evolution would make economic agents behaveas if they indeed maximized their payoffs in a world of scarce material resources in our view contains some grain of truth. The question is: Under which circumstances is behavior more of the ‘standard’ homo oeconomicus-type, and which environments defined by which conditions induce human beings to act e. g. like homo reciprocans (cp. Fehr and G¨achter 1998) and benevolent dictators?
This paper combines methods of evolutionary and behavioral game theory. We attempt to make progress on the above question about environmental and psycho-logical determinants of material-payoff maximization as opposed to social prefer-ences in a very simple model. We concentrate on a possible preference for reci-2The argument that stakes are simply too small for people to bother about monetary
pay-off maximization has been countered by experiments in which stakes were equivalent of several month’s income of participants (e. g. the experiments conducted in Indonesia by Cameron 1999 or in the Slovak Republic by Slonim and Roth 1998).
procity as formalized by Falk and Fischbacher (2001), in an environment consisting not of just one, but two distinct distribution tasks. As a first approximation of the complexities of the real ‘game of life’, we study evolution of reciprocity-based pref-erences in a world where agents randomly face either the ultimatum or the dictator game.3
The remainder of the paper is organized as follows: The following section clarifies the terms reciprocity and fairness, compares recent approaches to the definition of preferences that result in reciprocal behavior, and explains our choice of the Falk-Fischbacher formulation. Section 3 gives a brief introduction to the indirect evolutionary approach, section 4 presents our model, which is analyzed in section 5 with focus on the question: Under which circumstances is a reciprocity-based notion of fairness evolutionary stable? Section 6 compares results to those obtained in a related model by G¨uth and Napel (2002). Section 7 concludes.
2. RECIPROCITY
Positive (negative) reciprocityrefers to the impulse or desire to be kind (unkind) to those who have been kind (unkind) to us.4 Reciprocity is to be distinguished
from simple altruism, i. e. unconditional generosity. While the narrow self-interest hypothesis in ‘standard theory’ fails to explain stylized facts of many experiments, the notion of reciprocity preference sheds a fairly consistent light on possible moti-vational forces behind a number of observations. This section briefly points to major contributions to the literature which incorporate reciprocal behavior into standard game theory, starting with a stylized description of the experimental observations that motivated them.5
2.1. Evidence from Experiments
In experimentalultimatum games, most proposers offer between thirty and fifty percent of the available surplus (‘pie’) to the receiver while offers below twenty percent are rare.6 Small offers are usually rejected by responders, in which case
both players receive a zero share; the rejection rate decreases in the amount offered to the responder. An important observation is that responders more likely accept 3What we in the following refer to as a multi-game environment can also be viewed as a
recurrent version of a comprehensivesinglegame, in which Nature has the first move and selects either an ultimatum or dictator subgame.
4See Fehr and G¨achter (1998) for a more detailed account.
5Compare Klein (2000) for a detailed comparison of literature on reciprocity. Also see the
recent overview by Fehr and Schmidt (2002).
6Analysis of the ultimatum game was pioneered by G¨uth, Schmittberger, and Schwarze (1982).
For surveys on ultimatum games see Thaler (1988), G¨uth and Tietz (1990), Camerer and Thaler (1995) and Roth (1995).
lower offers when these are known to be made by a random device rather than a human player. In the relatedbest-shot game, the proposer can only choose between two payoff distributions – one very advantageous, one very disadvantageous to himself or herself. Here, participants usually accept a higher degree of inequity than in the ultimatum game.7 In thedictator game the second player is no longer
responding but must accept any offer by the first player. Proposers usually offer less than in ultimatum games. However, about 80 percent offer a positive amount.8
In thegift-exchange game the first mover (firm) offers a wage wto the second mover (worker). The worker responds by choosing a level of privately costly effort. Higher wages yield lower payoff for the firms and higher ones for workers, while higher effort levels have the opposite effect. The standard game-theoretic prediction based on individual maximization of payoffs is that workers invariably choose the lowest possible effort level. In anticipation of this, firms will only make the lowest possible wage offer. In contrast to these predictions, many experimental subjects as well as real firms offer wages above the market rate, and are rewarded by positive effort levels increasing in wage.9
Inpublic goods games agents simultaneously decide whether to pay for the pro-vision of a public good. Experiments show that the amount subjects contribute is increasing in their expectation about the contributions of others.10 Fehr and
G¨achter (2000) provide evidence that a significant fraction of subjects is willing to bear additional private costs if detected free-riders can be punished.
All these experimental results strongly suggest that subjects care about more than their own material payoff. Fairness considerations of one sort or the other seem important; analytical models with predictive value should take this into account. Different approaches to this are conceivable.
2.2. Equity-based Approaches
Fehr and Schmidt (1999) and Bolton and Ockenfels (2000) are the most promi-nent models of agents with preferences that exhibitinequity aversion. Agents may increase their utility by sacrificing their own material payoff if by doing so their payoff is closer to their counterparts’ payoffs. Preferences can depend on the entire payoff distribution, but not on anyintentions ascribed to other players.
In Fehr and Schmidt (1999), agents dislike inequitable outcomes, but less so 7The best-shot game was introduced by Harrison and Hirshleifer (1989) and Prasnikar and
Roth (1992). See also Falk, Fehr, and Fischbacher (2000).
8See, for example, Forsythe, Horowitz, Savin, and Sefton (1994) and Andreoni and Miller
(2002).
9See, for example, Berg, Dickhaut, and McCabe (1995), Falk and G¨achter (2002) or Fehr,
G¨achter, and Kirchsteiger (1996).
when these are to their own advantage. Despite its simplicity, many stylized facts can be explained by this assumption. The model is consistent with giving in dic-tator, gift-exchange and closely related trust games and with the rejection of low offers in ultimatum games. Since the model does not account for intentions, it fails to explain why people behave differently when playing against a random device instead of a real player, or why low offers in a best-shot game are more readily accepted than in an ultimatum game.
The approach by Bolton and Ockenfels (2000) is similar. Their model contains fewer assumptions about functional forms than that of Fehr and Schmidt and agents compare own material payoff to the group’s average, making the reference point for fairness considerations endogenous. Marginal disutility of small deviations from equality is assumed to be zero; so if subjects are non-satiated in their own material payoff they will never propose an equal split in the dictator game. The model can explain giving in dictator and gift-exchange games as well as rejections in ultimatum games. However, it fails to explain punishment patterns in public goods games. Neither can it explain behavior that depends on inequities between other players.11
2.3. Intention-based Approaches
A crucial feature of the psychology of reciprocity is that decisions to be kind or unkind to others are based not only onmaterial consequences implied by other players’ actions but also on theintentions attributed to these players. Agents who are motivated by reciprocity discriminate between players who take an (un)generous action by choice and those who are forced to do so.12
Prominent formal formalizations of reciprocity in terms of intentions have been given by Rabin (1993), Dufwenberg and Kirchsteiger (2001) and Falk and Fis-chbacher (2001). Rabin was the first to adopt the framework of psychological games of Geanakoplos, Pearce, and Stacchetti (1989) to model reciprocity. He introduced so-calledfairness games in which a reciprocity payoff is added to the material pay-off of the players. The reciprocity paypay-off is calculated as the product of akindness term and areciprocity term. The kindness term is positive whenever a player feels 11For example, Charness and Rabin (2002) let a playerC choose between the material payoff
allocations (xA, xB, xC) = (575,575,575) and (900,300,600), in both of which playerC receives his or her fair share of 1
3. Bolton and Ockenfels’ model predicts the second choice as the payoff
for playerCis higher, but 54% of subjects choose the first allocation in the experiments.
12Experimental evidence is given in Falk, Fehr, and Fischbacher (2003) for four different
mini-ultimatum games. In each game the proposer had two choices, one of which was always to offer 20%. The alternatives were 0%, 20% 50% or 80% respectively. The rejection rate of the 20% offer was highest when the alternative was equal division, it was lowest when the only alternative was to offer nothing to the second player. The fact that the rejection rate was not zero in this case suggests that pure equity considerations also play a role.
being treated well. Then he or she tries to make the reciprocity term positive, too, in order to increase his or her total utility payoff. This can be achieved by being nice in return, as the reciprocity term is defined to be positive when the player chooses a kind action. Negative reciprocity is modelled analogously.
Rabin’s original model is restricted to two-player normal form games. Dufwen-berg and Kirchsteiger (2001) generalize Rabin’s model ton-person extensive form games of imperfect information. Requiring that behavior is sequentially rational, the main complication is to keep track of the beliefs which determine the attributed intentions as the game evolves.
Falk and Fischbacher (2001) extend Rabin’s approach to perfect information extensive form games with finitely many stages. They construct a utility function that allows for pure equity concernand for intention-based reciprocity. This com-bination implies an additional degree of freedom by which – as is to be expected – more experimental observations can be explained than by either pure approach. Equilibrium calculations and parameter calibrations are already complex for en-tirely intention-based models of reciprocity;13this becomes yet more pressing with
Falk and Fischbacher’s hybrid model.
Both Falk and Fischbacher (2001) and Dufwenberg and Kirchsteiger (2001) pro-duce predictions more or less in line with the experimental observations mentioned above. For the ultimatum game, Dufwenberg and Kirchsteiger’s model unfortu-nately yields multiple sequential reciprocity equilibria (SRE). In all SREs high of-fers are accepted and low ofof-fers are rejected. However, there is no unique prediction for responder reaction to medium-range offers. For every parameter constellation there exists an SRE in which the proposer offers the smallest amount which is ac-ceptable to the responder. For high reciprocity parameters there also exist SREs in which equilibrium offers are rejected, because the players view one another as unkind, which in equilibrium, in fact, has both players act unkindly.
Falk and Fischbacher’s reciprocity equilibrium for the ultimatum game involves a mixed responder strategy. Acceptance probability is increasing in the offered amount up to a cut-off level, from which on the offer is always accepted. The cut-off is decreasing in the responder’s reciprocity parameter and always lower than one half. The optimal offer by a selfish proposer matches this level. If the proposer is reciprocal, too, his or her offer can but need not be more generous, depending on how reciprocal he or she is relative to the responder. The equilibrium in random-offer ultimatum games, in which the offer is not determined by a human but by a random device, highlights the role of intentions: Because no intentions are attributed to a random offer, the acceptance probability of a given offer is higher than in the regular ultimatum game. It is not necessarily one, because pure equity considerations also play a role.
The model of Falk and Fischbacher (2001) correctly predicts the main stylized facts for the gift-exchange game, too. Worker’s effort choice is increasing in wage paid and her reciprocity parameter. Strictly positive wages result. Predictions by the Falk and Fischbacher model for the best-shot game, the dictator game and the public goods game are similarly consistent with the experimental evidence discussed above. This and its comprehensiveness – covering pure inequity aversion or pure intentional reciprocity as special cases – makes it a very powerful tool for the analysis of many different games and our choice in the following analysis.
3. INDIRECT EVOLUTION
Theindirect evolutionary approach studies the evolution of preferences among rational decision makers. Agents are rational in the sense that they have prefer-ences and select strategies with the goal to maximize expected utility. Preferprefer-ences can but need not depend on an agent’s individual material payoff alone, they may e. g. exhibit reciprocity in the way discussed above. However, it is assumed that (average) material payoffs ultimately determine the fitness of agents having partic-ular preferences – and hence preferences themselves – in an evolutionary process: Agents with preferences that earn material payoffs above (below) the average re-produce more (less) successfully, and their and their preferences’ population share increase (decrease).
The considered agents make choices based on anticipated consequences, evaluat-ing their options by preferences which evolved dependent on past successes. Thus, as argued in more detail e. g. in Berninghaus, G¨uth, and Kliemt (2003) and G¨uth, Kliemt, and Napel (2003), the indirect evolutionary approach combines the main elements of the traditional neoclassical approach (rational purposeful actions se-lected according to their anticipated consequences or the ‘shadow of the future’) and the direct evolutionary approach (agents carry out fixed behavioral programs which evolve based only onpast success or the ‘shadow of the past’).14
Utility and material payoff are distinguished. Agents’ objectives – including possibly a preference for reciprocity – are described by their utility functions, which under additional conditions imply a unique equilibrium outcome for a given game. Equilibrium material payoffs are interpreted as fitness of the respective 14The difference between the direct and indirect evolutionary approaches is smaller than it
appears: Models that study the evolution of behavior can be regarded as the special case of preference evolution where feasible preferences are restricted to make a distinct feasible strategy strictly dominant. Models that study preference evolution in a game Γ can be regarded as studying direct evolution in a higher-level game Γ0. The latter can be constructed by identifying each player’s strategy setS0 with the possible preferences on outcomes of Γ and by taking the payoff functionπ0 to be the compositionπ◦µof the mappingµfrom strategy profilesS0N in Γ0 (i. e. a preference regarding outcomes of Γ for each playeri∈N) to equilibrium strategy profiles of Γ and the original payoff functionπ.
preference type. A number of studies has used this method to evaluate the sta-bility of different kinds of preferences in specific single-game environments. Huck and Oechssler (1999), for example, explore spiteful preferences in a mini-version of the ultimatum game. Ko¸ckesen, Ok, and Sethi (2000a) and Ko¸ckesen, Ok, and Sethi (2000b) analyze interdependent preferences in symmetric aggregative games and certain classes of symmetric supermodular and submodular games, respec-tively. Sethi and Somanathan (2001) consider reciprocal preferences in aggregative games. Berninghaus, G¨uth, and Kliemt (2003) study the trust game and Poulsen and Poulsen (2003) investigate reciprocity, altruism, and materialism in prisoner’s dilemma games.
Using the same utility function in different games, it is possible to calculate material success of general aspects of preferences in a complex environment. A particular sense of fairness (or absence thereof) seems to be such a general character trait rather than a game-specific preference. The indirect evolutionary approach extended to multiple games thus lends itself to stability tests of the preferences discussed in section 2.
The assumption of preferences for reciprocity has been justified above in terms of its success in making sense of and predicting behavior in experiments. Finding preferences so that they fit a given set of experimental results is useful in a de-scriptive sense. This may, however, be regarded as merely a question of sufficiently many free parameters and econometric exercise. We want to investigate if the as-sumption could be justified even without the data, just by answering the question: Which preferences within a given class – which is a restriction already – would one expect to prevail given the axiom that evolutionary success is monotonic in mate-rial payoffs? Evolutionary stability of reciprocity would in our view constitute an additional reason to update the common narrow view of homo oeconomicus as a purely self-interested materialist.
The following analysis is motivated by G¨uth and Napel (2002). There, equity-based preferences similar to those of Fehr and Schmidt (1999) are subjected to evolutionary pressures in an environment consisting of a highly stylized ‘game of life’. It does clearly not reflect the complexities of the real ‘game of life’ which has shaped the preferences of experimental subjects. However, it seems already a little bit closer to real social environments than any even more stylized single-game environment. Here, the multi-game indirect evolutionary analysis of G¨uth and Napel15is extended to the domain of psychological games, using preferences of the
Falk-Fischbacher type, which reflects not only inequity aversion but also intention-based reciprocity. We deem it interesting to see whether the way in which fairness preferences are modelled matters. A direct comparison with G¨uth and Napel’s results is contained in section 6.
4. THE MODEL
Repeatedly, two agents who are randomly drawn from a single population are given the chance to create a surplus (the ‘pie’), and subsequently to decide about its distribution in a single-shot game. Taking the population to be large enough to rule out repeated-game effects, agents are assumed to act fully rationally according to commonly known preferences for the possible outcomes of the single-shot game presented to them. Utility is not restricted to own material payoff, but it is the expected material payoff which defines an agent’sfitness orreproductive success in the evolutionary process imposed on the population.16 As expressed in a pointed
way by Samuelson (2001, p. 226f): “Nature can thus mislead her agents, in that preferences and fitness can diverge, but cannot mislead herself, in that high fitness wins the day.”
4.1. The Material World
A matched pair of agents faces either of two different games: The ultimatum game or the dictator game. In the ultimatum game one of the two players is randomly (with probability 0.5) selected to be the proposer (roleX), who proposes how to split the pie of one unit. The amount he or she offers is denoted byc∈[0,1]. The other player (the responder, role Y) decides whether to accept the proposed split or to reject it. The resulting material payoffs are (πX, πY) = (1−c, c) and
(πX, πY) = (0,0), respectively.
In thedictator game the agents are also assigned to be proposer or responder with equal probability. The proposer decides how to split the pie by offering c∈
[0,1] to the responder. In contrast to the ultimatum game the responder must accept, so resulting material payoffs are (πX, πY) = (1−c, c).
Which game the agents play is determined randomly, with probabilityλ∈[0,1] for the ultimatum game and 1−λ for the dictator game. The game realization becomes common knowledge. In both events material payoffs (πX, πY) determine
reproductive success.
Differences between dictator game versus ultimatum game very loosely resemble those between competitive market interaction versus private interaction: Success in the ultimatum game depends on mutual cooperation and is influenced by both agents’ behavior; in the dictator game one player is a mere price taker without influence on terms of trade. It will be interesting to see how evolutionary stable preferences depend on thefrequency or ‘importance’ of each type of interaction in agents’ lives.
16See Benaim and Weibull (2003) for a concise overview of models with agents who learn or
imitate rather than biologically reproduce which can be very closely approximated by evolutionary models coming from a purely biological background.
4.2. Agents’ Preferences
Agents have fairness preferences as defined by Falk and Fischbacher (2001). We prefer not to explicitly restate their quite complex utility function here, since we will directly work with the reciprocity equilibria calculated for dictator and ultimatum games by Falk an Fischbacher. Importantly, utility functionsuX
i (·) and
uY
i (·) which represent preferences of a given agent iin the roles of proposer X or
responderY, respectively, have the free parametersρX
i , εXi andρYi , εYi .17
Agents’ behavior depends on these free parameters. Most important is the reciprocity parameter ρk
i ∈R+, which describes how much weight agentiplaces on
reciprocal behavior in role k∈ {X, Y}. For ρk
i = 0 the considered agent is purely
interested in his or her ownmaterial payoff, behaving as in most orthodox economic models. Therejection probabilityof ultimatum game offers below 50% turns out to be strictly increasing in reciprocity parameterρY
i .
The second parameter, εk
i ∈ [0,1], measures agent i’s pure concern for an
eq-uitable outcome in role k. Agents without any pure equity concern (εX
i = εYi =
εi = 0) have an entirely intentions-driven notion of fairness. If no intentions can be
attributed to the opponent player, the considered agent maximizes material payoff. In contrast, for agents with εi >0 equitable splits of the pie are valuable even in
the absence of intentions. This is best seen in the dictator game where (anticipated or own) reciprocal behavior plays no role: An agent withεi= 0 always offersc= 0
whereas an agent withεi= 1 may offer up to half of the pie.18
In principle, preferences could evolverole-dependently (ρX
i 6=ρYi andεXi 6=εYi ),
so that an agent’s way of evaluating intentions and material consequences of choices depends on his or her role in the game. However, the beauty of the idea to outfit agents with a utility function which reflectsgeneralagent characteristics like equity concern or reciprocity preference is that agentsgenerally apply this utility function to the situations they face – regardless of the actual situation and regardless of theirrole in the situation.19 Therefore the parameters are assumed to evolve
role-independently in the following, i. e.ρX
i ≡ρYi ≡ρi andεXi ≡εYi ≡εi for any given
17Playeri’s utility at any terminal nodetof the game tree is the sum of his or her material payoff associated with tand a reciprocity payoff which is calculated decision node by decision node along the path totand weighted by parameterρi. For each ofi’s decision nodesnalong the path tot, the products of a term measuring the (un)kindness of others toiwhich has lead ton(here, parameterεimay matter) and a term measuringi’s reciprocated (un)kindness from choosing to move further towardstare added up. Equilibrium calculations are complex.
18The actual offer depends on the pure equity-concernandthe reciprocity parameters. 19The possibility that agents can condition equity concern on the game and / or their role in it is
explored in G¨uth and Napel (2002). If real economic agents’ fair behavior is adequately described by a utility function with one or a few fairness-related parameters at all, the same parameters should be valid for more than a special class of games. To expect them to be valid inallgames seems, however, too much.
0.1 0.2 0.3 0.4 c 1 2 3 4 5 ΡY 0 0.25 0.5 0.75 1 p÷ 0.1 0.2 0.3 0.4 c
FIG. 1Acceptance probabilityp∗ of agent with reciprocity parameterρ
Y for offercin
the ultimatum game
agent. Note that, moreover, the same two parametersρi andεi will be applied to
both ultimatum and dictator games, i. e. the social part of agent i’s preferences is assumed to not depend on the realized type of game.
5. EVOLUTIONARY ANALYSIS
First, ultimatum and the dictator games are studied separately, corresponding to the cases λ = 1 and λ = 0. Then, the mixed environment consisting of both games is analyzed. When agents A and B interact, we will in the following write
ρX (ρY) for that agent’s (role-independent) reciprocity parameter who is assigned
to roleX (roleY).
5.1. The Ultimatum Game Equilibrium play20 results in acceptance probability
p∗= minn1, c ρY·(1−2c)(1−c) o if c < 1 2 1 if c≥1 2 (1) for offercconditional on the responder’s reciprocity parameterρY 6= 0, andp∗ = 1
forρY = 0. An offer of half of the pie or more is always accepted for sure. This is not
the case for lower offers, for which the acceptance probability is strictly decreasing in ρY (see figure 1).
The share c∗ offered by the proposer X to the responder Y in equilibrium is
20The computations are given in detail in Falk and Fischbacher (2001, appendix 3). Cf. foot-note 17 above.
1 2 3 4 5 ΡX 1 2 3 4 5 ΡY 0 0.1 0.2 0.3 0.4 0.5 c÷ 1 2 3 4 5 ΡX 0 0.1 0.2 0.3 0.4 0.5 c÷
FIG. 2 Splitc∗ proposed by an agent with reciprocity parameter ρ
X to one with
reci-procity parameterρY in the ultimatum game
given by c∗= maxn1 + 3ρY − p 1 + 6ρY +ρ2Y 4ρY ,1 2 ·(1− 1 ρX )o (2)
forρX, ρY 6= 0. ForρX > ρY = 0 andρY > ρX = 0,c∗ is (2)’s limit forρY ↓0 and
ρX ↓ 0, respectively, while forρX =ρY = 0 one obtainsc∗= 0 (corresponding to
‘traditional’ preferences).
The equilibrium sharec∗ is defined by two expressions. The first depends only
on the responder’s reciprocal inclination ρY and is the maximizer of proposer’s
expected utility in case that the responder’s concern for reciprocity is ‘binding’ – dominating the insufficiently pronounced reciprocity concern of the proposer. The associated share of the pie is just high enough to ensure acceptance (p= 1). The second expression depends only on the proposer’s ρX and reflects how much the
proposer wouldvoluntarilyoffer in order to maximize utility in the light of his or her intrinsic concern for a fair outcome. As eitherρX orρY grow large,c∗ approaches
1/2 (see figure 2).
The equilibrium offer is the maximum of both expressions. This means that if a selfish proposer plays against a reciprocal responder, the offer is increasing inρY.
If the responder’s concern for reciprocity is low, the offer depends on the fairness concern of the proposer, hence it is increasing inρX. If both players are selfish, the
offer is close to zero.
AgentA’s role in interaction with agent B is random, with probability 0.5 for either role. The expected material payoffπ∗
AU of agentAwith preference parameters (ρA, εA) who is matched with agent B to play the ultimatum game is the average
0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.3 0.4 0.5 0.6 ΠA÷ U 0.5 1 1.5 2 2.5 ΡA 0.3 0.4 0.5 0.6 ΠA÷ U
FIG. 3Expected payoffπ∗
AU of agentAwith reciprocity parameterρAin the ultimatum game when matched against an agentB with reciprocity parameterρB
π∗ AU(ρA, ρB) = 5ρA+3− √ 1+6ρA+ρ2A 8ρA if ρA≥ −2ρB ρB+1− √ 1+6ρB+ρ2B (a) 3ρB−3+ √ 1+6ρB+ρ2B 8ρB if ρA≤ ρB(ρB−1) 1+ρB (b) 3ρA+1− √ 1+6ρA+ρ2A 8ρA + ρB−1+ √ 1+6ρB+ρ2B 8ρB if −2ρB ρB+1− √ 1+6ρB+ρ2B > ρA> ρB1+(ρBρB−1). (c) (3) Figure 3 illustrates this. Expected payoff depends on both agents’ concerns for reciprocity, captured byρAand ρB.
There are three cases. In case (a),ρAis large compared toρBand thus behavior
in both possible role realizations is determined by agent A’s strong concern for fairness. Agent A’s payoff is strictly decreasing in ρA. So evolution would drive
ρ down until case (c) applies: The population share of agents with payoffs below average (with a higherρ) will decrease and the share of agents with a payoff above average (with a lower ρ) will increase.21
Case (b) applies ifρA is small compared to ρB. Then, behavior in both roles
is determined byρB, and agentA’s payoff is constant inρA. Given the symmetry
of our setup – both agents are drawn from thesame population – this is similar to 21We are intentionally a bit sloppy in describing evolutionary dynamics. In order to be precise
we would have to define a payoff-monotonic selection dynamic which is either started from an interior initial state (in which every ρis present) or is augmented by an innovative mutation process which ensures that everyρgets a chance to spread. We refrain from a formal treatment because the technicalities of properly dealing with the state space ∆(R+) consisting of probability
distributionsµon the nonnegative real line are immense and do not add anything to our main argument. See Oechssler and Riedel (2002) for a flavor of the mathematical techniques that would be necessary.
case (a), but now from agent B’s point of view: Payoff is higher for lower ρB so
that evolution again drives ρdown until case (c) applies.
In the intermediate case (c),ρA andρB do not differ too much. The proposer
(regardless of whether agent A or B is given this role) offers a split that maxi-mizes his or her own material payoff subject to the binding constraint imposed by the rejection behavior of the responder, i. e.c∗ depends only on the responder’s
reciprocity parameter. Then agentA’s (B’s) payoff is increasing inρA(ρB).
Evolu-tion results inρin the entire population approaching infinity within the variability bound defining case (c). This is in the following referred to asρ‘diverging’.
Whatever the initial distribution of the reciprocity parameter in the population is, ρwill diverge. For the ultimatum game this implies anequal split in the limit. It is in the nature of this split, that it is always offered for strategic reasons, i. e. to prevent the responder from rejecting, not because of the proposer’s intrinsic motivation.
5.2. The Dictator Game
The split c∗ offered by the proposer X in the dictator game in equilibrium
depends only on his or herownreciprocity parameterρXand pure outcome-concern
parameterεX: c∗= maxn0,1 2·(1− 1 εXρX) o . (4)
A proposer offers a positive amount ifεXρX>1, which means that he or she is
reasonably concerned about reciprocity in the unintentional case. No matter how largeεXρX is, the offer is never greater than half. Equation (4) corresponds to the
second term in (2), i. e. the possible ‘voluntary offer’ in the ultimatum game, where
εXρX replaces ρX. Here, the receiver has no choice but to accept and therefore
the outcome is ‘unintentional’ in the sense of Falk and Fischbacher (2001); this is reflected by a ‘discounting’ of reciprocity parameterρX by pure equity concernεX
in the dictator game. A given agent always offers weaklyless in the dictator game than in the ultimatum game, because 0≤εX ≤1.
AgentAplays the dictator game with agentB and is assigned the proposer role with probability 0.5. His or her expected material payoffπ∗
AD in equilibrium is then given by:
0.5 1 1.5 2 2.5 ¶AΡA 0.5 1 1.5 2 2.5 ¶BΡB 0.3 0.4 0.5 0.6 ΠA÷ D 0.5 1 1.5 2 2.5 ¶AΡA 0.3 0.4 0.5 0.6 ΠA÷ D
FIG. 4Expected payoffπ∗
AD of agentAwith parametersεAandρAin the dictator game when matched against an agentBwith parametersεB andρB
π∗AD(εA, ρA, εB, ρB) = 1 2 if εAρA≤1 andεBρB ≤1 (I) 1 4+4εA1ρA if εAρA>1 andεBρB ≤1 (II) 3 4−4εB1ρB if εAρA≤1 andεBρB >1 (III) 1 2+4εA1ρA − 1 4εBρB if εAρA>1 andεBρB >1. (IV) (5) We have four cases, illustrated by figure 4. In cases (II) and (III) either of the two agents offers a positive amount to the other agent when he or she is in the role of proposer. The expected payoff for such ‘generous’ behavior is smaller than in case (I), and strictly decreasing in the agent’s parameters ρandε. Cases (II) and (III) are hence unstable – evolution will shift preferences towards case (I).
In case (IV)both agents as proposers share the pie with the receiver; their payoff need not be smaller than in case (I). Still, material payoff is decreasing in each agent’s individual parameters ρ and ε, so again a payoff-monotonic evolutionary process drives agents’ preferences towards case (I).
For parameter constellations in case (I) both agents are only weakly interested in fairness; both offerc∗ = 0 as proposers. The case applies as long as the product of
the parameters for reciprocityρand the pure equity-concernεis smaller or equal to unity for both agents. All such parameter constellations are behaviorally equivalent, so that any of them corresponds to an evolutionary stable state.22 Constellations
falling into case (I) are the only stable ones.
This implies that in the long run agents offer zero in the dictator game and 22Here, we refer to stability in the spirit of neutrally stable strategies and Lyapunov dynamic
Λ=0.5,¶A=0.5,¶B=0.5 0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.4 0.45 0.5 0.55 0.5 1 1.5 2 2.5 ΡA 0.4 0.45 0.5 0.55 Λ=0.5,¶A®0,¶B®0 0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.4 0.45 0.5 0.55 0.5 1 1.5 2 2.5 ΡA 0.4 0.45 0.5 0.55 Λ=0.5,¶A=1,¶B=1 0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.4 0.45 0.5 0.55 0.5 1 1.5 2 2.5 ΡA 0.4 0.45 0.5 0.55 Λ=0.5,¶A=0.5,¶B=1 0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.4 0.45 0.5 0.55 0.5 1 1.5 2 2.5 ΡA 0.4 0.45 0.5 0.55
FIG. 5 Expected payoffπ∗
A of agentAwith parameters εA andρA in the ‘game of life’ when matched against an agentBwith parametersεB andρB forλ= 0.5
thus cannot be distinguished from selfish payoff maximizers. Still, the model allows agents to have a preference for fairness up to a level that does not affect behavior in the dictator game. The ‘upper bound’ on this hidden fairness is given byρ·ε≤1; any combination of concern for reciprocity ρ and pure concern for equity ε that satisfies this condition can persist in the population if only the dictator game is played.
5.3. The Mixed Environment
Let us now consider the stylized ‘game of life’ in which agents get randomly involved either in a ultimatum or a dictator game. The payoffπ∗
Aof agentAfacing
agentBgiven the probabilityλ∈[0,1] to play the ultimatum game and probability 1−λto play the dictator game is
π∗
A(εA, ρA, εB, ρB, λ) =λ·π∗AU(ρA, ρB) + (1−λ)·π
∗
AD(εA, ρA, εB, ρB). (6) This expected material payoff function is illustrated forλ= 1/2 and several values of εA and εB in figure 5.23 Total expected payoff is increasing in payoff π∗AU(·) from the ultimatum game and the payoff π∗
AD(·) from the dictator game. Hence 23Note the saddle points (1,1) and (2,2) in the cases ofε
A =εB = 1 and εA = εB = 0.5, respectively, and that a saddle point – corresponding to an evolutionary stable state if εwere fixed (or is restricted as in section 5.4) – does not exist forεA=εB≈0.
evolution would favor preferences that maximize π∗
AU(·) and π
∗
AD(·) at the same time if this were possible.
Indeed, this is possible for the Falk-Fischbacher preferences considered here: The maximum payoff in the dictator game is reached for any parameter constellation withρ·ε≤1. The pure equity concern parameterεcan always totally compensate a high reciprocity concern ρ by being smaller or equal to 1/ρ. So, the ‘optimal’, material payoff-maximizing preferences in the dictator game can be reached for any level ofρ. In the light of our earlier analysis the evolutionary limit outcome in the mixed environment is hence a diverging reciprocity parameter ρ and convergence of the pure equity concern parameter εto zero such thatρ·ε≤1.
In the limit, agents behave very fairly whenever their counterpart can recipro-cate, but show no pure concern for equity at all. The long-run results are equal splits in the ultimatum game and full appropriation of the pie in the dictator game, i. e. we obtain a simple superimposition of the long-run outcomes in the separate com-ponents of our stylized ‘game of life’. With parameterρdiverging andεconverging towards zero, agents reciprocate strongly in situations that imply intentionality but do not care at all about fairness when it comes to situations where intentions play no role.
5.4. Restricted Parameters in the Mixed Environment
With two free parameters and two games, it is not surprising – though neither trivial – that evolution brings about ‘optimal’ behavior for isolated dictator and ultimatum game environments also in the combined habitat. While our agents are unable to have different sets of parameters for reciprocity and pure equity concern in ultimatum and dictator game, respectively, i. e. they cannot discriminate directly between the games, they succeed toindirectlydiscriminate based on distinctive fea-tures of the games (here: intentionality). It can be questioned whether the degrees of freedom in the social component of our preferences in reality match or exceed the different classes of social interaction.24 Humans seem to adjust behavior to a
specific situation only to some extent. It therefore seems interesting and reasonable to limit nature’s freedom in shaping agents’ preferences25 by restricting the
possi-ble range of the pure outcome-concern parameter ε, which applies in unintentional situations.
One practical possibility is to impose an exogenous lower bound εl > 0 for
parameter ε, so thatεl ≤ ε≤1 is required instead of 0 ≤ ε≤1. This situation
24Exogenous costs of discrimination by a given agent’s (global) preferences would probably play
a central role in a theoretical investigation of this relationship.
25The available preferences are already limited by the specific family of utility functions chosen
by us. See Dekel, Ely, and Yilankaya (1998) for a model with finite strategy space and no such restriction.
Λ=0.7,¶A=¶B=¶l=0.5 0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.4 0.5 0.6 0.5 1 1.5 2 2.5 ΡA 0.4 0.5 0.6 Λ=0.9,¶A=¶B=¶l=0.5 0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.4 0.5 0.6 0.5 1 1.5 2 2.5 ΡA 0.4 0.5 0.6 Λ=0.3,¶A=¶B=¶l=0.5 0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.4 0.5 0.6 0.5 1 1.5 2 2.5 ΡA 0.4 0.5 0.6 Λ=0.5,¶A=¶B=¶l=0.5 0.5 1 1.5 2 2.5 ΡA 0.5 1 1.5 2 2.5 ΡB 0.4 0.5 0.6 0.5 1 1.5 2 2.5 ΡA 0.4 0.5 0.6
FIG. 6 Expected payoffπ∗
A of agent A with parameters εA = εl = 0.5 andρA in the ‘game of life’ when matched against an agentB with parametersεB =εl= 0.5 andρB for several values of λ
differs from the above in that a strong concern for reciprocity cannot completely be blanked out in the dictator game by a low ε. An agent’s behavior in dictator-realizations of the ‘game of life’ is thus no longer de facto independent from that in ultimatum-realizations.
Given the boundεl, the dictator game continues to induce persistent downward
pressure on parameterεforρ≥1/εl. Therefore, forρ≥1/εlthe evolutionary stable
outcome involves ε = εl. At the same time the ultimatum game puts persistent
upward pressure on reciprocity parameter ρ. This has the following implications forρ:
For small values ofλ, the dictator game is ‘important’ enough in agents’ lives to implyρ·εl= 1 orρ= 1/εl. This results in a zero offer in the dictator game and
c= 1 + 3 1 εl− q 1 + 61 εl + 1 ε2 l 41 εl (7) in the ultimatum game. For example, a value of εl = 0.25 implies a split of
(πX, πY) ≈(0.59,0.41) or εl = 0.125 implies (πX, πY) ≈ (0.55,0.45) in the
ulti-matum game in the long run. The imposed restriction thus slightly reduces eq-uitableness in ultimatum offers without consequences for selfish behavior in the dictator game.
0.2 0.4 0.6 0.8 1 ¶l 0.825 0.85 0.875 0.9 0.925 0.95 0.975 1 Λ0
FIG. 7 The critical valueλ0(εl) at which ρ= 1/εlceases to be an evolutionary stable
point
point at (ρA, ρB) = (2,2), corresponding to a stable state withρ= 1/εl, is barely
visible for λ= 0.7 and no longer exists for λ= 0.9. In fact, there is a critical level of λ above which (ρA, ρB) = (1/εl,1/εl) is no longer stable and ρ diverges. This
critical level can be calculated by analyzing expected payoff for values ofρ >1/εl;
if it is lower (higher) than for ρ = 1/εl the latter combination is (is not) stable.
Constellationsρ >1/εl belong to case (c) in the ultimatum game and case (IV) –
involving positive offers by both agents – in the dictator game. The material payoff of agent Ain the ‘game of life’ is therefore
π∗ A(εA, ρA, εB, ρB, λ) = λ " 3ρA+ 1− p 1 + 6ρA+ρ2A 8ρA +ρB−1 + p 1 + 6ρB+ρ2B 8ρB # +(1−λ) · 1 2+ 1 4εAρA − 1 4εBρB ¸ . (8)
The marginal payoff to agentA from greaterρAis
∂π∗ A(εA, ρA, εB, ρB, λ) ∂ρA = εAλ(1 + 3ρA) + (2λ−2−ρAλ) p 1 + 6ρA+ρ2A 8εAρA p 1 + 6ρA+ρ2A (9) and can be analyzed forρA>1/εlandεA=εlin order to find out at which critical
levelλ0 the combination of selfish dictator offers and (moderately) fair ultimatum
offers becomes unstable. A’s marginal payoff is negative for values of λup to the critical level λ0(εl) = 2q1 + 6 εl + 1 ε2 l 3 +εl+ (2−εl) q 1 +ε6 l + 1 ε2 l (10) and then positive. Forλ≥λ0(εl) evolution results in diverging parameterρbecause
increasing in ρA. As illustrated in figure 7, λ0(εl) is strictly decreasing in the
exogenous lower bound on pure equity concern εl.26
So when the ultimatum game is playedvery frequently compared to the dictator game (λ≥λ0), the reciprocity parameterρdiverges and the pure outcome concern
resides atε=εl. This implies equal splits in both games in the long run.27 Loosely
speaking, if agents interact in comparatively many ‘private’ situations captured by the ultimatum game compared to few ‘market’ situations reflected by the dictator game, they develop a strong notion of fairness.
6. COMPARISON WITH G ¨UTH AND NAPEL (2002)
Recall that the crucial difference between this model and that of G¨uth and Napel (2002) is the considered class of preferences. Agents’ sense of fairness is limited to inequity aversion in the paper of G¨uth and Napel, while here a partly intention-based preference for reciprocation is investigated.
Results of the separate analysis of ultimatum and dictator game are qualitatively the same. In theultimatum game agents in the role of the responder benefit from inequity aversion as well as from reciprocation, because the proposer anticipates that a small offer might be rejected. Evolution leads agents who exclusively play the ultimatum game to such high levels of inequity aversion or reciprocation, that in the long run a rather equitable or even equal split is reached (this depends on the precise utility specification).
In thedictator game, in contrast, both inequity aversion and reciprocal behavior are detrimental to an agent’s average success, as it leads him or her to voluntarily give part of the pie away. Still, in G¨uth and Napel’s investigation as well as this one, agents can be inequity averse to any degree that fails to actually affect proposer behavior. In the analysis by G¨uth and Napel, this upper bound to stable inequity aversion of dictators is driving results for the mixed environment. Either the long-run level of inequity aversion in the ultimatum game is below this bound, resulting in no behaviorally relevant interaction between the two games, or it is above. In the latter case, the given composition of the stylized ‘game of life’ determines a stable level that, loosely speaking, balances marginal benefits and costs of inequity aversion. In our analysis the situation is more complicated due to the presence of two free parameters instead of one. The original model of Falk and Fischbacher (2001) allows for both: Strong reciprocation and no inequity aversion in uninten-tional cases at all. If the possible discrepancy between these two aspects of fairness is not restricted exogenously, the reciprocity parameters goes to infinity and agents
26It falls fromλ
0= 1 forεl−→0 to a minimum ofλ0= 2 √
2−2≈0.838 forεl= 1.
27A diverging parameterρimplies equal split in the ultimatum and dictator game as long as
parameterεdoes not converge towards zero. For an explanation see section 5.1 on the ultimatum game and section 5.2 on the dictator game.
behave in both games just as if they were independent from each other. This cor-responds to G¨uth and Napel’s case in which agents can explicitly condition the social component of their utility function – and hence fair behavior – on the game at hand. Here, such a moral discrimination between games is achieved indirectly. When ‘discounting’ of fairness in theunintentional case is limited, the outcome re-sembles qualitatively the one of G¨uth and Napel’s main case of game-independent equity aversion.
It is a common result of either analysis that a high share of ultimatum games affects evolutionary stable preferences. G¨uth and Napel find moderate increases in the stable level of inequity aversion after raising λ, while here the reciprocity parameter diverges once a critical point is reached. Coincidentally, the minimal level at which the amount of ultimatum games played affects the outcome is similar in both studies, about 80% in G¨uth and Napel (2002) and around 85% for a wide range of constellations here; below these thresholds, the contrasting long-run outcomes for dictator and ultimatum games are independent of the precise composition of the stylized ‘game of life’. In summary, both specifications of the utility function evolution lead to qualitatively similar outcomes in ultimatum, dictator and mixed environments. The impact that an explicit inclusion of reciprocal behavior has on the evolutionary stable outcome is small in the considered class of games if the flexibility increase from introducing an additional parameter is limited.
7. CONCLUDING REMARKS
In general, psychological fairness models explain experimental results better than more tractable equity-based models, which in turn perform better than the default assumption of monetary payoff maximization. This success comes at the ex-pense of additional free parameters, which have do be defended against accusations of ad hoc ‘game fitting’, and often more complicated analysis with sometimes am-biguous results. Since parameterscan be such that an agent, in fact, has traditional materialistic preferences, indirect evolutionary analysis provides a useful theoretical benchmark. If under plausible modelling assumptions evolutionary forces selected parameters that imply non-degenerate fairness preferences, this would corroborate the post-experimental econometric estimations.
The relevance of fairness preference a priori can be expected to vary with the decision situations at hand even if these are restricted to simple distribution tasks. Our analysis suggests that preference for fair behavior has sound evolutionary rea-sons, but – in line with experimental observations – more likely plays a significant role in non-degenerate games with punishment opportunities, such as the ultima-tum game, than in what is basically a one-player decision problem like the dictator game. If Nature permits players to condition the social component of their
pref-erences on different games (as investigated by G¨uth and Napel 2002) or at least indirectly on differences in games, the same agents can exhibit a pronounced sense of fairness in one type of social interaction while they are entirely selfish in another one. This is true even though preferences are assumed to evolve simultaneously in a multi-game environment.
If Nature imposes physical or psychological restrictions on the variance in agents’ social attitudes across different games or different dimensions of fairness, the pre-cise composition of the stylized ‘game of life’ faced by agents has an impact. In our model, evolutionary pressure towards more reciprocal preferences created by the ultimatum game is strong enough to induce quite equitable behavior (offers above 40%) in this game independently of how frequently it is played. In contrast, the success of evolutionary pressure towards more selfish preferences created by the dictator game is sensitive to the share of dictatorial decision problems: If agents most of the time face the more fairness-encouraging ultimatum game, they will eventually feel it in their interest to be generous even as dictators. Whether this asymmetry – a great share of fairness-encouraging games eliminates selfish behav-ior in materialism-encouraging games, while the reverse is not true – in response to possible changes in the composition of the environment is a more general phe-nomenon or an artifact of our (and also G¨uth and Napel’s) model specification remains to be seen.
We have studied environmental and psychological determinants of material-payoff maximization vs. social preferences in a particularly simple two-game envi-ronment. In our view, this is an improvement compared to the usual analysis of preferences in a single game. It highlights the importance of possibly unconscious links between behavioral modes in different classes of interaction. Whether human agents indeed face a binding restriction in their ability to discriminate between the fairness implications of similar play in ultimatum and dictator games, reflected by the reciprocity and pure equity concern parameters ρ and ε in our model, is an empirical question. Experimental evidence on the ultimatum game is consistent with both the unrestricted (section 5.3) and restricted evolution of ρand ε (sec-tion 5.4); evidence on the dictator game supports the restricted view, though not our prediction of equal splits.28
It seems desirable to us to extend our ‘game of life’ to more than only ultimatum and dictator games. First, an environment with more than two component games would create a natural endogenous restriction on the two parameters of the con-sidered Falk-Fischbacher reciprocal preferences – preventing the full ‘specialization’ observed in section 5.3 without exogenous restrictions as in section 5.4. Second, reciprocal preferences are proposed as explanations for empirical observations of 28Dictator offers in G¨uth and Napel (2002) continuously rise from zero to moderate positive
a considerable range of games, including e. g. trust and gift-exchange games, pub-lic good games, and variants of the ultimatum game such as the best-shot game. Analysis of the stability of reciprocal preferences should include these.
Another very desirable extension would be to relax our a priori restriction of the class of feasible preferences to those of the Falk-Fischbacher type.29 These
include the neoclassical reference point of selfish materialism as well as salient other special cases (e. g. purely intention-based fairness or a general equity concern). They admittedly exclude many other types of agents, which are competing with the traditional homo oeconomicus for economists’ attention, too.
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SONDERFORSCHUNGSBereich
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Martin Hellwig
A Utilitarian Approach to the Provision and Pricing
of Excludable Public Goods
03-35
Daniel Schunk
The Pennsylvania Reemployment Bonus
Experiments: How a survival model helps in the
analysis of the data
03-34
Volker Stock´e
Bettina Langfeldt
Umfrageeinstellung und Umfrageerfahrung. Die
relative Bedeutung unterschiedlicher Aspekte der
Interviewerfahrung f¨ur die generalisierte
Umfrageeinstellung
03-33
Volker Stock´e
Measuring Information Accessibility and Predicting
Response-Effects: The Validity of
Response-Certainties and Response-Latencies
03-32
Siegfried K. Berninghaus
Christian Korth
Stefan Napel
Reciprocity - an indirect evolutionary analysis
03-31
Peter Albrecht
Cemil Kantar
Random Walk oder Mean Reversion? Eine
statistische Analyse des Kurs/Gewinn-Verh¨altnisses
f¨ur den deutschen Aktienmarkt
03-30
J¨urgen Eichberger
David Kelsey
Burkhard Schipper
Ambiguity and Social Interaction
03-29
Ulrich Schmidt
Alexander Zimper
Security And Potential Level Preferences With
Thresholds
SONDERFORSCHUNGSBereich
504
WORKING PAPER SERIES
Nr.
Author
Title
2
03-28
Alexander Zimper
Uniqueness Conditions for Point-Rationalizable
Solutions of Games with Metrizable Strategy Sets
03-27
J¨urgen Eichberger
David Kelsey
Sequential Two-Player Games with Ambiguity
03-26
Alain Chateauneuf
J¨urgen Eichberger
Simon Grant
A Simple Axiomatization and Constructive
Representation Proof for Choquet Expected Utility
03-25
Volker Stock´e
Informationsverf¨ugbarkeit und Response-Effects:
Die Prognose von Einfl¨ussen unterschiedlich
kategorisierter Antwortskalen durch
Antwortsicherheiten und Antwortlatenzen
03-24
Volker Stock´e
Entstehungsbedingungen von Antwortverzerrungen
durch soziale Erw¨unschtheit. Ein Vergleich der
Prognosen der Rational-Choice Theorie und des
Modells der Frame-Selektion
03-23
Daniel Schunk
Modeling the Use of Nonrenewable Resources
Using a Genetic Algorithm
03-22
Brian Deal
Daniel Schunk
Spatial Dynamic Modeling and Urban Land Use
Transformation: An Ecological Simulation
Approach to Assessing the Costs of Urban Sprawl
03-21
Thomas Gschwend
Franz Urban Pappi
Stimmensplitting und Koalitionswahl
03-20
Thomas Langer
Martin Weber
Does Binding or Feeback Influence Myopic Loss
Aversion - An Experimental Analysis
03-19
Peter Albrecht
Carsten Weber
Asset/Liability Management of German Life
Insurance Companies: A Value-at-Risk Approach
in the Presence of Interest Rate Guarantees
03-18
Markus Glaser
Online Broker Investors: Demographic
Information, Investment Strategy,
